A Channel-Adaptive Range-Doppler Domain Filtering Serial BAQ Algorithm and Comparative Analysis

Abstract

With the growing demand for large-scale urban observation, multi-channel technology has become a cornerstone of high-resolution wide-swath SAR systems. The challenge of storing and transmitting the large data volumes generated by multi-channel systems has driven the development of advanced data compression techniques. However, in onboard implementations with non-power-of-two channel numbers and serial data formats, the existing multi-channel compression algorithms reveal significant conflicts involving channel counts, FFT cores, and the Krieger method. To address these issues, this paper introduces the Channel-Adaptive Range-Doppler domain filtering Serial Block Adaptive Quantization algorithm (CARDS-BAQ). By incorporating a point-frequency RD domain filtering approach and leveraging serial data matrix splicing and rollback combined with point-frequency ABAQ, CARDS-BAQ enables efficient data compression for arbitrary channel counts. The performance of CARDS-BAQ is validated using GF-3 measured data through comparative analysis with BAQ, ABAQ, MCBAQ, and 3MBAQ algorithms under power-of-two channel conditions. Additionally, its applicability and reliability for non-power-of-two channel numbers are demonstrated through payload flight experiments conducted in 2024 in Yingkou, Liaoning Province, China. CARDS-BAQ effectively supports data storage and transmission for large-scale urban observation, marking a significant advancement in remote sensing technology.

1. Introduction

Synthetic aperture radar (SAR) offers the capability for all-weather and all-day observation, making it highly valuable for applications such as resource surveys, military reconnaissance, and disaster monitoring. To enhance the range and detail of urban target information, SAR systems are evolving rapidly towards achieving higher resolution and wider swaths. However, achieving a wide ground observation band requires a low pulse repetition frequency (PRF), while attaining a high azimuth resolution demands a sufficiently large PRF [1]. This creates a conflict between swath and resolution requirements in single-channel SAR systems.

Azimuth multi-channel SAR (MC-SAR) effectively mitigates Doppler spectrum ambiguity by coherently combining echo signals received in the azimuth direction, enabling the achievement of high resolution and wide swath under low-PRF conditions [2]. However, while this approach enhances observational capabilities, it also imposes significant demands on hardware design. The data volume collected by MC-SAR systems is several times greater than that of earlier systems, leading to stringent requirements for storage and transmission bandwidth in spaceborne SAR systems. Advanced data compression techniques are essential to alleviate these pressures, either by achieving a higher compression ratio for equivalent data or image quality, or by preserving more image details within limited storage capacities.

Numerous SAR raw data compression algorithms have been developed over time [3,4]. In the time domain [5], scalar quantization algorithms [6,7] and vector quantization algorithms [8,9,10] leverage the statistical characteristics and intrinsic correlation of the raw data. In the transform domain [11,12], techniques such as wavelet transform compression, sub-band coding, and FFT transform compression [13,14] offer improved performance but are computationally more complex. Other methods include lattice coding quantization [15], predictive coding [16,17,18], compressed sensing [19,20], and entropy limit quantization [21,22]. Among these, block adaptive quantization (BAQ) has become the mainstream method for SAR data compression due to its algorithmic simplicity and ease of hardware implementation [6,23]. However, the widespread adoption of BAQ does not guarantee optimal performance across all SAR radar types. For spaceborne multi-channel SAR systems, as discussed in the following section, BAQ and similar methods fall short. On one hand, few existing algorithms propose joint data compression schemes specifically for azimuth multi-channel scenarios. On the other hand, most existing algorithms remain at the theoretical stage, with limited analysis of hardware implementation. As a result, the current azimuth multi-channel data compression algorithms are fundamentally unable to be implemented on airborne or spaceborne systems with a non-power-of-two number of channels.

This paper addresses these issues by proposing a Channel-Adaptive Range-Doppler domain filtering serial BAQ algorithm (CARDS-BAQ). The algorithm introduces a point-frequency filtering method in the RD domain tailored for input azimuth channels with non-power-of-two configurations. Combined with matrix splicing and rollback processing of serial data and single-frequency-point adaptive bit allocation BAQ (ABAQ), the algorithm achieves high-performance channel-universal data compression.

The structure of this paper is organized as follows. Section 2 reviews the existing compression algorithms prior to CARDS-BAQ to lay the groundwork for subsequent comparisons. Section 3 outlines the algorithmic flow and key components of CARDS-BAQ. In Section 4, the algorithm performance is evaluated through both real-world data and engineering validation, using BAQ, ABAQ, MCBAQ, and 3MBAQ as baselines to verify the effectiveness of CARDS-BAQ under arbitrary channel configurations. Section 5 provides a brief discussion and interpretation of the test results. Finally, Section 6 summarizes the work, highlighting the key advantages, limitations, and future directions of CARDS-BAQ.

2. Related Work

2.1. Block Adaptive Quantization

BAQ [24] is the most widely used method for spaceborne SAR data compression. This algorithm offers several advantages, including a simple principle, ease of hardware implementation, effective compression performance, and fast computation speed. The BAQ algorithm achieves effective data compression by controlling the optimal quantizer [25] based on the standard deviation of the input data for each sub-block. Furthermore, BAQ allows for a further reduction in computational complexity by exploiting the correspondence between the amplitude mean and standard deviation [6,10,26].

Despite the maturity and widespread adoption of BAQ techniques [27,28,29,30,31], several limitations remain. Firstly, under real-time processing constraints, the resource demands of BAQ scale proportionally with the number of channels, and increasing the number of sampling points leads to a significant rise in hardware resources such as accumulators. Secondly, in azimuthal multi-channel SAR systems, conventional BAQ compression does not leverage the inherent inter-channel correlations to further reduce the compression ratio.

2.2. Adaptive Bit Allocation Block Adaptive Quantization

Adaptive bit allocation BAQ (A-BAQ) algorithms have emerged to accommodate variations in terrain scattering characteristics. Notable examples include Flexible Dynamic BAQ (FDBAQ) [32] and ABAQ [3], which adaptively select the optimal quantization bit depth based on the signal amplitude of individual data blocks without relying on prior knowledge of the terrain scattering properties in specific regions [33].

The ABAQ data compression approach presents two key limitations. Firstly, it fails to exploit the inherent correlations among channels in multi-channel SAR data. Secondly, as ABAQ operates in the time domain, it encounters difficulties in uniformly scattering target regions where the standard deviation across sub-blocks exhibits minimal variation, thereby rendering the adaptive bit allocation scheme ineffective. Furthermore, under small-offset conditions, the algorithm’s performance is significantly affected by the bit error control mechanism [34].

2.3. Multi-Channel Block Adaptive Quantization

Multi-channel BAQ (MCBAQ) [35] transfers raw SAR data from the time domain to the transform domain via a Discrete Fourier Transform (DFT) across channels, enabling adaptive bit allocation in the transform domain based on spectral energy levels to enhance compression efficiency. To resolve the issue of fractional bit allocation, the method incorporates Azimuthal Switching Quantization (ASQ) [36]. BAQ is then applied across sub-blocks as the final stage of compression.

MCBAQ demonstrates clear advantages by incorporating inter-channel coupling in the azimuth dimension [37,38,39]. Through the application of a DFT across channels, it effectively mitigates non-uniform sampling issues in slow time and enables adaptive bit allocation across the frequency spectrum. Nevertheless, the method has certain limitations. Firstly, ASQ enables fractional bit rate control but does not account for inter-block dependencies based on the statistical characteristics of the data. Secondly, procedures such as integration and matrix transposition pose significant challenges for onboard implementation. Thirdly, when dealing with wide azimuth bandwidths, the spectral energy may become concentrated in a limited number of frequency bins, resulting in suboptimal bit allocation.

2.4. Multi-Channel Multi-Pulse Multi-Weight Block Adaptive Quantization

Figure 1 illustrates the data compression flow of the 3MBAQ algorithm for N-channel SAR echo data. More specifically, 3MBAQ acquires multi-channel raw data from a high-resolution ADC and reconstructs an equivalent single-channel Doppler domain data matrix using the Krieger reconstruction algorithm [2,40,41,42]. This matrix is partitioned into several sub-bands, where non-uniform bit allocation is determined based on multi-weighted contributions. Subsequently, A-BAQ compression is performed within each sub-band matrix, as detailed in Equations (1)–(3).

$$P_k={\displaystyle \frac{1}{N_a·N_r}}\sum _{n=1}^{N_r}\sum _{m=1}^{N_a}{\left|y_k[m,n]\right|}^2$$

(1)

$$I_k={\int }_{-PBW/2}^{PBW/2}{\left|{\displaystyle \frac{sin\left(\pi N\frac{f+PRF(N/2-n)/N}{PRF}\right)}{sin\left(\pi \frac{f+PRF(N/2-n)/N}{PRF}\right)}}\right|}^{2}df$$

(2)

$${\sigma }_k^2={\displaystyle \frac{P_k\times I_k}{N^2}}$$

(3)

where ${\sigma }_n^2$ represents the weighted ‘equivalent variance’, $P_k$ denotes the signal energy, and $I_k$ is the integral of the Dirichlet kernel [24] over the effective bandwidth. $y_k[m,n]$ denotes the Doppler distance-domain data for the kth frequency band corresponding to position $(m,n)$, and $1/N^2$ serves as a scale-variation factor and does not contribute in the actual bit allocation. $N_r$ and $N_a$ represent the number of sampling points in the range and azimuth directions for a single frequency band, respectively. N denotes the number of azimuth channels, and PRF stands for the pulse repetition frequency.

Further, 3MBAQ effectively mitigates the FFT noise floor issue and overcomes the spectral resolution limitations of MCBAQ in scenarios with a small number of channels. Nonetheless, its onboard implementation presents several challenges, such as the constraint that the number of channels must be a power of two, and the presence of stitching errors arising from block-wise processing.

3. Materials and Methods

Based on the advantages and disadvantages of BAQ, ABAQ, MCBAQ, and 3MBAQ, a Channel-Adaptive Range-Doppler domain filtering serial BAQ algorithm (CARDS-BAQ) is proposed. This algorithm is capable of adapting to different numbers of channels and introduces an azimuth multi-channel RD filtering method for non-power-of-two channels, achieving azimuth signal reconstruction and aggregation. Furthermore, under the condition of serial input, a partitioning and splicing method is proposed to enable high-performance data compression of multi-channel SAR in the transform domain, effectively reducing the onboard data storage and transmission pressure. The flowchart of the algorithm is shown in Figure 2.

3.1. Channel Adaptation

The Krieger filter [2,42] requires that the number of FFT frequency points must be divisible by the number of channels shown in Equation (4).

$${\displaystyle \frac{N_{FFT}}{N}}=k$$

(4)

where $N_{FFT}=2^n,n,k\in \mathbb{N}$ is a requirement for engineering implementation.

In general, the number of azimuth channels N in high-resolution wide-swath SAR systems is not a power of 2. Due to hardware and Vivado’s FFT IP limitations, the number of FFT frequency points $N_{FFT}$ must be a power of 2, which does not satisfy the requirements of Krieger’s azimuth signal reconstruction. This issue is often overlooked during algorithm design and simulation as it deviates from the underlying logic of hardware implementation. CARDS-BAQ proposes an RD filtering method that adapts to the input number of channels.

Under the condition where the number of input channels is N and $N_{FFT}$ is assumed to be a power of 2, one data record unit corresponds to one azimuth channel. The number of azimuth PRTs (Pulse Repetition Times) input in a single cycle of the data record unit is $\lfloor {\displaystyle \frac{N_{FFT}}{N}}\rfloor$. Each channel’s azimuth pulses are zero-padded to $N\times \lfloor {\displaystyle \frac{N_{FFT}}{N}}\rfloor$ intervals, and an additional $N_{FFT}-N\times \lfloor {\displaystyle \frac{N_{FFT}}{N}}\rfloor$ zero pulses are appended at the end of the azimuth pulses. The resulting data matrix has azimuth pulses, where $\lfloor {\displaystyle \frac{N_{FFT}}{N}}\rfloor$ pulses are valid data and the remaining are zero pulses.

In cases where the number of channels N is not a power of two, the Krieger filter fails to achieve uniform spectral division. To overcome this limitation, CARDS-BAQ proposes a point-frequency Range-Doppler (RD) filter, with the corresponding filter function $g\left(f\right)$ defined in Equations (5)–(7).

$$g_n\left(f\right)=G_{nk}\left(f^{\prime }\right),k=⌈{\displaystyle \frac{f}{PRF}}+{\displaystyle \frac{N}{2}}⌉$$

(5)

$$\begin{array}{cc}\hfill G\left(f\right)& =H^{-1}\left(f^{\prime }\right)\hfill \\ \hfill & =\left[\begin{array}{ccc}{H}_1(f^{\prime };\Delta x_1)& \cdots & H_N(f^{\prime };\Delta x_N)\\ ⋮& \ddots & ⋮\\ H_1(f^{\prime }+(N-1)\times PRF;\Delta x_1)& \cdots & H_N(f^{\prime }+(N-1)\times PRF;\Delta x_N)\end{array}\right]\hfill \\ \hfill & =\left[\begin{array}{ccc}{G}_{11}\left(f^{\prime }\right)& \cdots & G_{1N}\left(f^{\prime }\right)\\ ⋮& \ddots & ⋮\\ G_{N1}\left(f^{\prime }\right)& \cdots & G_{NN}\left(f^{\prime }\right)\end{array}\right]\hfill \end{array}$$

(6)

$$H_n(f;\Delta x_n)=exp\left(-j{\displaystyle \frac{\pi \Delta x_n^2}{2\lambda r_0}}\right)exp\left(-j2\pi {\displaystyle \frac{\Delta x_n}{2v}}f\right)$$

(7)

where f represents the point frequency under $N_{FFT}$-point Fourier transformation, and $g_n\left(f\right)$ is an RD filter proposed by CARDS-BAQ, corresponding to channel n. $\Delta x_n$ is the distance between the phase center of the transmitting antenna and the n-th receiving antenna, v is the flight velocity of the platform, $\lambda$ is the carrier wavelength, and $r_0$ is the nearest slant range. The pulse repetition frequency under N-channel azimuth is PRF, and $f\in \left[-{\displaystyle \frac{RRF\times N}{2}},{\displaystyle \frac{PRF\times N}{2}}\right]$ represents the frequency range indicated by a single channel.

The point-frequency filter $g_n\left(f\right)$ is obtained by solving a reconstruction filter matrix $G\left(f^{\prime }\right)$, where the independent variable $f^{\prime }$ of the matrix must be located at the leftmost part of the spectrum corresponding to the equivalent pulse repetition frequency, i.e., $-{\displaystyle \frac{RRF\times N}{2}}\le f<-{\displaystyle \frac{PRF\times (N-2)}{2}}$. The relationship between $f^{\prime }$ and f is given by Equation (8).

$$f^{\prime }=f-⌊{\displaystyle \frac{f}{PRF}}+{\displaystyle \frac{N}{2}}⌋$$

(8)

From this, a range-Doppler domain filter for any number of channels can be obtained. This filter can determine the corresponding filtering factors for any frequency point, enabling azimuth multi-channel signal reconstruction.

The processing flow for signal reconstruction using the point-frequency filter is shown in Figure 3. Here, $h_m(t,r_0)$ represents the echo signal response function for a single channel, $A_1\left(vt\right)$ is the amplitude modulation function in channel 1, and ${\tilde{A}}_{RX}(·)$ and ${\tilde{A}}_{TX}(·)$ are the single-channel amplitude modulation functions. $\Delta \varphi$ denotes the multi-channel phase offset, and $\Delta t$ represents the multi-channel time delay. By controlling the sampling frequency through the PRF signal, the equivalent multi-channel signal is obtained.

3.2. Serial Input Data

Taking the input data format into account, the use of serial data input results in the received data being structured as small matrices at the processing end. Building upon the assumptions from the previous subsection, the input to the matrix splicing stage consists of a Range-Doppler (RD) matrix comprising $N_{FFT}$ azimuth pulses. In the theoretical context of limited data volume, splicing issues caused by data partitioning are not considered. However, when the input format shifts to continuous input, addressing the splicing problem after data reconstruction becomes a necessary requirement. CARDS-BAQ proposes a card-like overlapping data splicing method. By rolling back and replaying data from the data recording unit, the RD matrix of azimuth pulses is inverse Fourier transformed (IFFT) back to the time domain, with both ends truncated to retain the valid data with minimal error. The process flow is illustrated in Figure 4.

As shown in Figure 4, $N_{shift}$ represents the number of data steps for each rollback playback. The number of azimuth pulses played each time, $N_{azimuth}$, is given by $\lfloor {\displaystyle \frac{N_{FFT}}{N}}\rfloor$. The final retained data matrix’s azimuth pulse count is still a power of 2, that is, $N_{valid}={\displaystyle \frac{N_{FFT}}{2}}$. It is important to note that

(1) The subsequent single-point frequency ABAQ needs to be performed in the frequency domain. ${\displaystyle \frac{N_{FFT}}{2}}$ is the largest power of 2 smaller than $N_{FFT}$. This allows for reducing concatenation errors by trimming the data matrix and ensuring the maximum amount of data are processed each time.

(2) To ensure that information is retained without aliasing, the truncation of data is performed in the time domain after the point-frequency filter. The truncation is conducted by keeping the ${\displaystyle \frac{N_{FFT}}{2}}$ pulses in the middle of the $N_{FFT}$ pulse range during each truncation operation.

(3) Aliasing is avoided by rolling back and playing back pulse data from the original data. Each channel plays $\lfloor {\displaystyle \frac{N_{FFT}}{N}}\rfloor$ pulses at a time. After applying zero-padding and FFT operations, the equivalent single-channel data following point-frequency filtering contain $N_{FFT}$ pulses. After matrix cropping and stitching, the number of pulses in the resulting RD matrix is ${\displaystyle \frac{N_{FFT}}{2}}$. To ensure the data volume bandwidth is as closely matched as possible, the step size for each data rollback, $N_{shift}$, can be obtained from the following Equation (9). Each playback involves a certain repetition of data pulses. This ensures that the final data are free from aliasing.

$$N_{shift}=⌊{\displaystyle \frac{N_{FFT}}{2N}}⌋$$

(9)

(4) Under the condition where the number of channels is not a power of 2, the data processing bandwidth $\lfloor {\displaystyle \frac{N_{FFT}}{2N}}\rfloor \times N\ne {\displaystyle \frac{N_{FFT}}{2}}$ is mismatched. This requires obtaining the data at the ground station and restoring them to the time domain, after which the ${\displaystyle \frac{N_{FFT}}{2}}-\lfloor {\displaystyle \frac{N_{FFT}}{2N}}\rfloor \times N$ pulses at the end must be discarded to achieve aliasing-free stitching. The overall data volume difference is

$$V_{diff}={\displaystyle \frac{N_{FFT}-\lfloor N_{FFT}/2/N\rfloor \times 2N}{N_{FFT}}}$$

(10)

When the number of channels is a power of 2, the data volume difference is 0, enabling downward compatibility for systems with a power-of-two number of channels.

After completing the second spectral transformation, the input data are an RD matrix with ${\displaystyle \frac{N_{FFT}}{2}}$ azimuth pulses. The theoretical basis for data compression in the frequency domain is provided in the following subsection. Additionally, the statistical properties of each input matrix are updated using a sliding averaging method.

Taking the SAR image processed directly from the three-channel data matrix without segmentation as the baseline, the image data processed after dividing the large matrix into smaller matrices along the azimuth direction is shown in Figure 5. It exhibits periodic large errors in the azimuth direction. If the large matrix is divided into smaller equally sized matrices with partial overlap, resulting in a greater number of smaller matrices, and the overlapping PRT portions are removed after processing, the periodic errors caused by splicing can be effectively reduced, as shown in Figure 6.

3.3. Single-Frequency-Point ABAQ

While SAR raw data generally conform to an approximately zero-mean Gaussian distribution, they exhibit relatively low correlations in both the range and azimuth directions. Orthogonal transformation enables decorrelation by converting highly correlated spatial samples into uncorrelated or weakly correlated transform coefficients, thus eliminating redundancy inherent in the data. As demonstrated in [34], for the Discrete Fourier Transform (DFT), optimal bit allocation and quantization can be effectively carried out solely on the transform coefficients.

When the number of FFT points ($N_{FFT}$) is small and the spectral information has already been focused in the RD domain, conventional BAQ block partitioning may not be suitable as each block could encompass most of the spectral energy. To address this, a single-frequency-point ABAQ partitioning scheme is proposed, in which each sub-block has a Doppler-domain length of one. Theoretically, this fine-grained partitioning aligns well with the concentrated spectral distribution. From a hardware perspective, single-frequency-point partitioning avoids an increase in the number of accumulators. Specifically, it eliminates the need for additional accumulators that would otherwise be required to temporarily store data in multi-frequency-point sub-blocks due to the priority of range-direction data input.

Furthermore, this single-frequency-point design offers two additional hardware advantages. On the hand, it enables the use of statistical characteristics from the previous pulse in the RD domain to quickly compress the current input for adjacent pulses. On the other hand, compared to multi-frequency-point partitioning, single-frequency-point partitioning provides statistical characteristics that are more aligned with the current input, resulting in smaller errors. The optimal solution for non-uniform bit allocation using single-frequency-point ABAQ in the Range-Doppler domain is given by Equations (11) and (12).

$$\begin{array}{cc}\hfill \underset{R_n=0,1,2,\cdots ,N-1}{argmin}J& =\underset{R_n=0,1,2,\cdots ,N-1}{argmin}\left[\overline{d}+\lambda \left({\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{R}_n-\overline{R}\right)\right]\hfill \\ \hfill & =\underset{R_n=0,1,2,\cdots ,N-1}{argmin}\left[{\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{d}_n+\lambda \left({\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{R}_n-\overline{R}\right)\right]\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill R_n& =\overline{R}+lo{g}_2{\sigma }_n-{\displaystyle \frac{1}{N}}\sum _{m=0}^{N-1}lo{g}_2{\sigma }_m\hfill \\ \hfill & =\overline{R}+\Delta R_n-R_{avg}\hfill \end{array}$$

(12)

where N is the total number of sub-blocks, $\Delta R_n$ represents the bit offset corresponding to the standard deviation of the n-th sub-block, $R_{avg}$ is the average bit offset across all sub-blocks; $\overline{d}\left(\overline{R}\right)$ denotes the average distortion, ${\sigma }_n^2$ represents the source variance in the SAR raw data domain, $R_n$ is the number of quantization bits for the n-th sub-block, $d_n$ denotes the distortion measure of the sub-block, and $\overline{R}$ is the predefined average quantization bit rate.

4. Results

4.1. GF-3 Data Test Results

Given that the CARD algorithm operates under the same principle for any power-of-two number of channels, the simulation in this subsection uses GF-3 data with two azimuth channels to assess the performance of CARDS-BAQ. In the subsequent subsection, flight experiments are conducted under a test scenario involving a non-power-of-two channel configuration. The flight parameters of the GF-3 dataset used in this subsection are provided in

Table 1. Partial parameters of GF-3 data.

Parameter	Value
Number of azimuth channels, N	2
Frequency band	C
Azimuth antenna length, $L_{az}$	7.5 m
Pulse repetition frequency, $PR{F}_{sys}$	2179 Hz
Total processed bandwidth, $PBW$	2466 Hz
Minimum slant range, $r_0$	902,273 m
Effective radar velocity, $v_{sat}$	7133 m/s
Range sample frequency, $f_s$	133 MHz
Range bandwidth, $B_r$	120 MHz

.

The data compression performance is evaluated using Signal-to-Quantization-Noise Ratio (SQNR) and Normalized Mean Square Error (NMSE) as metrics in Equations (13) and (14), with BAQ, ABAQ, MCBAQ, and 3MBAQ serving as comparative baseline algorithms. The metric results of the five algorithms applied to GF-3 data under varying compression ratios are presented in

Table 2. Performance of 5 data compression algorithms at different compression ratios on GF-3 data.

Index	Ratio	BAQ	ABAQ	MCBAQ	3MBAQ	CARDS-BAQ
SQNR	8:1	4.4518	4.4518	4.9234	5.6423	6.4982
	8:2	9.4566	9.4566	9.6458	10.7385	11.3476
	8:3	14.8848	14.8848	14.7778	15.6083	16.3055
	8:4	20.5168	20.5168	20.2248	20.5788	21.3546
NMSE	8:1	0.3588	0.3588	0.3219	0.2728	0.0224
	8:2	0.1133	0.1133	0.1085	0.0844	0.0733
	8:3	0.0325	0.0325	0.0333	0.0275	0.0234
	8:4	0.0089	0.0089	0.0095	0.0088	0.0073

.

$$SQNR=10lo{g}_{10}{\displaystyle \frac{{\sum }_{i=1}^{N_a}{\sum }_{j=1}^{N_r}{\left|s_{ij}\right|}^2}{{\sum }_{i=1}^{N_a}{\sum }_{j=1}^{N_r}{|s_{ij}-{\tilde{s}}_{ij}|}^2}}$$

(13)

$$NMSE={\displaystyle \frac{{\sum }_{i=1}^{N_a}{\sum }_{j=1}^{N_r}{|s_{ij}-{\tilde{s}}_{ij}|}^2}{{\sum }_{i=1}^{N_a}{\sum }_{j=1}^{N_r}{\left|s_{ij}\right|}^2}}$$

(14)

where $s_{ij}$ and ${\tilde{s}}_{ij}$ represent the values of the uncompressed signal and compressed signal at point $(i,j)$, respectively. $N_a$ and $N_r$ denote the number of sampling points in the azimuth and range directions.

To provide a clear visualization of the performance trends and comparative analysis among the five algorithms, curves of SQNR and NMSE as functions of bit rate under various compression schemes are presented. It should be noted that BAQ and ABAQ yield identical performance at integer bit rates. However, since BAQ lacks the capability to support non-integer bit rate compression, it is not plotted separately in Figure 7.

From the perspective of image-domain analysis, Figure 8 presents a comparative visualization consisting of a 3D noise map, an image illustrating overall compression performance across the entire scene, and a detailed view of strong-point-target representation.

To demonstrate the compression performance of the algorithms at various bit rates, Figure 9 presents a comparison of 3D quantization noise maps in the image domain, obtained after compressing and reconstructing the data using BAQ/ABAQ, MCBAQ, 3MBAQ, and CARDS-BAQ at 1-, 3-, and 4-bit levels.

4.2. Project Flight Data Test Results

4.2.1. Equipment Used

To assess the performance of CARDS-BAQ in processing serial input data under non-power-of-two channel scenarios, the algorithm was implemented on a digital hardware platform using Vivado.

Table 3. Hardware and software environments of CARDS-BAQ.

Parameter	Value
Project device part	xc7vx690tffg1927-2
DDR3 model	UniIC-HXI15H4G160AF-13K
Amount of SDRAM	9
Internal organization	8 banks × 32 M bits × 16
Vivado edition	2018.1
Windows edition	Win10
System clock	100 MHz

lists the hardware specifications and software development environment, while

Table 4. Resource utilization of FPGA implementation on V7 development board.

Resource	Utilization	Available	Utilization %
LUT	223,646	433,200	51.63
LUTRAM	34,466	174,200	19.79
FF	318,634	866,400	36.78
BRAM	1221.5	1470	83.10
DSP	1682	3600	46.72

presents the associated hardware resource utilization.

The onboard implementation scheme is encapsulated in the data compression standalone system shown in Figure 10 and was tested in an azimuth three-channel flight experiment on the payload platform shown in Figure 11.

4.2.2. Flight Data

The payload platform of a wide-swath SAR project conducted a flight test in Yingkou, Liaoning Province, China, in 2024. The flight parameters are provided in

Table 5. Partial parameters of the wide-swath SAR project data.

Parameter	Value
Number of azimuth channels, N	3
Frequency band	X
Azimuth antenna length, $L_{az}$	1.5 m
Pulse repetition frequency, $PR{F}_{sys}$	1000 Hz
Minimum slant range, $r_0$	15,000 m
Effective radar velocity, $v_{sat}$	124 m/s
Range sample frequency, $f_s$	800 MHz
Range bandwidth, $B_r$	600 MHz

.

4.2.3. Flight Experiment Results

Table 6. FPGA performance based on ABAQ compression in spaceborne multi-channel SAR Doppler domain.

	Compared with Overall Reconstruction			Compared with Segmentation Reconstruction
	Complex	I	Q	Complex	I	Q
Ratio 1	≥8:2
Ratio 2	8:1.9826
SQNR (dB)	10.7653	10.7699	10.7607	10.9047	10.9093	10.9001
NMSE	0.0838	0.0838	0.0839	0.0812	0.0811	0.0813

presents the performance of the Channel-Adaptive Range-Doppler domain filtering serial BAQ algorithm as validated on the standalone system. Ratio 1 denotes the required compression ratio, whereas Ratio 2 corresponds to the compression ratio achieved in the implemented system.

Following the download and decompression of the compressed data, post-processing allows for the extraction of sub-block quantization bit counts and standard deviation information, as shown in Figure 12. Figure 13 presents a comparison between the image reconstructed from decompressed data during standalone testing and that from the original data, indicating that the critical detail information remains well preserved.

To further assess the performance of the standalone system integrated with CARDS-BAQ in real flight conditions, a flight test was carried out in 2024 utilizing the payload platform of the wide-swath SAR project in Yingkou, Liaoning, China. The resulting image from the acquired flight data is presented in Figure 14.

A strong point target within the imaging region of Figure 14 is selected for detailed analysis, as illustrated in Figure 15a. The image obtained after 32× interpolation is shown in Figure 15b. To evaluate the impact of CARDS-BAQ on imaging quality, the two-dimensional range and azimuth profiles of the selected point are depicted in Figure 16. The corresponding metrics—including Peak Side Lobe Ratio (PSLR), Integrated Side Lobe Ratio (ISLR), and Impulse Response Width (IRW)—before and after compression are summarized in

Table 7. Comparison of imaging quality in range and azimuth directions.

	Azimuth			Range
	ORIGIN	CARDS-BAQ	LOSS	ORIGIN	CARDS-BAQ	LOSS
PSLR (dB)	−21.02	−20.71	0.31	−16.13	−15.97	0.16
ISLR (dB)	−18.67	−17.73	0.84	−14.51	−14.02	0.49
IRW (m)	0.558	0.554	0.004	0.263	0.263	0

.

5. Discussion

5.1. Simulation-Based Comparative Analysis

A higher SQNR and a lower NMSE indicate lower quantization noise and better algorithmic performance. As shown in Table 2, CARDS-BAQ exhibits superior compression performance compared to advanced methods such as 3MBAQ across four different compression ratios. For example, at a compression ratio of 8:2 (with the original data bit width being 8 and the compressed data bit width being 2), CARDS-BAQ achieves an SQNR of 11.3476 and an NMSE of 0.0733. This trend is also evident in Figure 7, where, under uniform scene conditions, CARDS-BAQ consistently outperforms BAQ, ABAQ, MCBAQ, and 3MBAQ across all the tested compression levels. In particular, at the 8:2 compression ratio, CARDS-BAQ achieves a 1.9 dB improvement in SQNR over BAQ.

From the following Figure 8, it can be seen that CARDS-BAQ performs better than BAQ, ABAQ, MCBAQ, and 3MBAQ in the image domain. It can be observed that CARDS-BAQ produces the least quantization noise among all the compared algorithms, as shown in Figure 8d. From the image comparisons in Figure 8i–l, it is clear that, for strong point targets, CARDS-BAQ increases the quantization gap between the peak point and the surrounding uncorrelated information. Therefore, under a limited image bit depth, the noise energy around the strong point target is lower, and the signal-to-noise ratio is higher. The image-domain results in Figure 8 validate that CARDS-BAQ outperforms the existing advanced algorithms such as 3MBAQ in terms of compression performance.

In addition, we compared the 3D plots of quantization noise generated in the image domain after data compression and restoration of BAQ, ABAQ, MCBAQ, 3MBAQ, and CARDS-BAQ at 1 bit, 3 bits, and 4 bits, as shown in Figure 9. From this, it can be seen that, when we compare different compression ratios of the same compression method vertically, the quantization noise gradually decreases as the final quantization bit depth increases, as shown in Figure 9a,e,i. When we compare the performance of different methods at the same compression ratio horizontally, it is clear that CARDS-BAQ exhibits the best quantization performance among all the schemes, while the performance of 3MBAQ, MCBAQ, and ABAQ gradually declines. BAQ and ABAQ have the poorest compression performance, as shown in Figure 9i–l.

5.2. Experimental Performance Analysis

As depicted in Table 6, under the condition where the compression ratio requirement is ≥8:2, CARDS-BAQ ultimately achieves an average compression bit rate of 1.9826 bits. Since the original data are multi-channel, to compare pre- and post-compression data in the data domain, the algorithm must separately implement both the overall reconstruction and segmented reconstruction of the multi-channel data matrix to obtain comparable data matrix results.

The compressed data are compared with both the overall reconstructed data and the segmented reconstructed data, resulting in an SQNR (complex) of 10.7653 dB and 10.9047 dB and an NMSE (complex) of 0.0838 and 0.0812, respectively. The reasons for the SQNR and NMSE performance being inferior to the theoretical values of CARDS-BAQ shown in Table 2 are as follows:

1. The system’s compression ratio requirement strictly enforces a ≥8:2 condition. The actual bit rate of 1.9826 bits inevitably leads to the SQNR and NMSE performance being slightly worse than the values at 2 bits.

2. Due to the limitations of the Vivado project and the implementation on the digital board, certain steps produce results with some distortion. For instance, the two FFT operations and division operations, which rely on IP core outputs, introduce a degree of error within an acceptable range compared to theoretical algorithm calculations. This results in some degradation in the performance of the decompressed data.

Nevertheless, the performance of the CARDS-BAQ algorithm still surpasses that of traditional BAQ algorithms, validating its adaptability to non-power-of-two channel numbers and serial input data.

In Figure 12a, it is evident that the algorithm achieves high-performance quantization results by adaptively allocating quantization bit widths between data sub-blocks. Comparing Figure 12a,b, it can be observed that the adaptive allocation is based on the standard deviation information, with the quantization bit width being positively correlated with the signal amplitude.

As shown in Figure 14, CARDS-BAQ demonstrates excellent performance in the image domain. Due to the direct onboard processing of non-power-of-two channel data, azimuth multi-channel errors introduce noticeable blurring in the image domain. After applying multi-channel error correction, the result in Figure 14b shows that CARDS-BAQ maintains good compression performance. Furthermore, the analysis of the strong point target in Figure 15 and Figure 16 and Table 7 indicates that the maximum degradation in PSLR and ISLR is 0.84 dB, with a negligible impact on mainlobe broadening. The slight performance degradation is mainly attributed to quantization noise, which remains within an acceptable range. These results further validate the effectiveness, generality, and superiority of the proposed Channel-Adaptive Range-Doppler domain filtering serial BAQ algorithm.

5.3. Computational Complexity

From the perspective of implementation, assuming the number of channels is N, each channel provides $N_a\times N_r$ (azimuth × range) data during each compression. Because the flight parameters are fixed, the filter coefficient of the point-frequency filter is also fixed, which can be directly allocated from the upper computer, and the filtering process does not account for the complexity. During the reconstruction process, each range line adds N times the zero-padding FFT process for $N_a\times N=N_{am}$ points. For each frequency point, the filtering process requires one addition operation and three complex multiplications (i.e., $3\times 3=9$ real multiplications), resulting in $N_r\times N_{am}$ points of Doppler range domain data. The computational complexity of CARDS-BAQ mainly increases due to the zero-padding FFT process $O\left(N_r{N}_{am}lo{g}_2{N}_{am}\right)$.

Under the same conditions, the computational complexity of 3MBAQ, as presented in [34], is also $O\left(N_r{N}_{am}lo{g}_2{N}_{am}\right)$. According to [35], the computational complexity of MCBAQ is $O\left(N_r{N}_{am}lo{g}_{2}N\right)$. Since its DFT process is performed across sampling points of each channel at any given moment, its computational complexity is lower than that of 3MBAQ and CARDS-BAQ but still within the same order of magnitude. The above complexity analysis adds extra complexity to ABAQ’s computation, and the computational complexity of ABAQ is at the same level as BAQ, which is $O\left(N_r{N}_{am}\right)$.

In summary, the computational complexity of CARDS-BAQ, 3MBAQ, and MCBAQ is $O\left(N^{2}lo{g}_{2}N\right)$, which is slightly higher than the computational complexity $O\left(N^2\right)$ of BAQ and ABAQ but still within an acceptable range.

5.4. Resource Consumption

CARDS-BAQ compensates for the periodic stitching errors of decompressed data through matrix cropping and stitching, as shown in Figure 5. Matrix cropping only requires specifying the corresponding valid bit identifiers. The stitching operation does not consume any resources and is saved in frames to be transmitted to the ground station for stitching. Rollback playback saves additional resource overhead at the cost of algorithm processing speed. The storage resources consumed by CARDS-BAQ are primarily used for transposing the data matrix in DDR3. Under azimuth multi-pulse conditions, the custom DDR3 SDRAM control interface and simple dual-port RAM are used to perform the transpose, FFT, and IFFT. Except for the transpose, all the other stages are completed in the form of data streams without occupying RAM storage resources. Fixed system parameters determine the fixed point-frequency filter, and its filter coefficients are provided by the host computer, occupying a small amount of ROM resources. The filtering process of CARDS-BAQ is implemented through serial data streams, allowing resource reuse and only requiring a small increase in the number of multipliers and adders.

MCBAQ cannot be directly implemented under a non-power-of-two number of channels or requires more complex methods, leading to further resource consumption. For ease of comparison, it is assumed that the resource consumption analysis is conducted under a power-of-two number of channels. On one hand, MCBAQ lacks a mechanism to address stitching errors in decompressed data. On the other hand, although it uses only one inter-channel FFT, saving the storage resources required for transposition, it introduces significant computational complexity when performing sub-band quantization bit allocation due to the need for operations such as integration. This step consumes a substantial amount of DSP resources as a trade-off.

The resources occupied by BAQ are mainly in accumulators and registers, while ABAQ consumes more RAM and ROM resources for adaptive quantization bit allocation. For both algorithms, the resource usage is directly proportional to the number of channels, with a proportionality coefficient of 1. Doubling the number of channels doubles the resource consumption. In multi-channel scenarios, their resource usage is comparable to that of CARDS-BAQ.

As shown in Table 4, the resource utilization of CARDS-BAQ on a v7 digital board under a three-channel background for the wide-swath SAR project is presented. CARDS-BAQ involves extensive data storage and processing, resulting in relatively high usage of BRAM, DSP, and FF resources. Additionally, the adaptive quantization requires pre-storing all the possible quantization results for the 1-bit to 5-bit range, leading to relatively high usage of LUT and LUTRAM resources. Overall, the resources occupied by the CARDS-BAQ algorithm in hardware design and implementation are within an acceptable range.

In summary, compared to other data compression algorithms, CARDS-BAQ consumes more storage resources, primarily for the transposition process. MCBAQ consumes more computational resources, mainly for integration calculations. Both algorithms use similar amounts of resources in the subsequent ABAQ quantization stage, with overall resource consumption at the same order of magnitude. BAQ and ABAQ have similar resource consumption, which is proportional to the number of channels and, in multi-channel conditions, is comparable to the resource consumption of CARDS-BAQ.

6. Conclusions

In high-resolution wide-swath SAR systems with non-power-of-two azimuth channel configurations, the existing multi-channel data compression algorithms face implementation challenges due to conflicts between the FFT kernel and the Krieger reconstruction method with the number of channels. To address this limitation, this paper proposes a Channel-Adaptive Range-Doppler domain filtering serial BAQ algorithm (CARDS-BAQ). Built on Krieger’s azimuth signal reconstruction framework, the algorithm introduces a point-frequency filtering method in the RD domain, enabling seamless adaptation to non-power-of-two channel configurations. Additionally, it employs a data splicing and rollback approach to correct periodic PRT data errors arising from serial input. The final step—adaptive bit allocation for data compression—is performed using single-frequency-point ABAQ in the RD domain.

The superiority of CARDS-BAQ over BAQ, ABAQ, MCBAQ, and 3MBAQ was validated using GF-3 experimental data. At a compression ratio of 8:2, CARDS-BAQ achieved improvements of 1.9 dB in SQNR and 0.04 in NMSE compared to BAQ. Furthermore, this paper marks a significant breakthrough in transitioning CARDS-BAQ from theory to practical application. Hardware implementation in the wide-swath SAR project demonstrated the algorithm’s adaptability under non-power-of-two azimuth channel configurations and serial input conditions. The computational complexity of CARDS-BAQ is $O\left(N^{2}log\left(N\right)\right)$. By combining theoretical validation with engineering verification, this study confirms the performance superiority and adaptability of the CARDS-BAQ algorithm. It effectively supports data storage and transmission for large-scale urban observation in remote sensing systems with arbitrary channel configurations.